nondestructive resolution of higher harmonics of light...

VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

32

Nondestructive Resolution of Higher Harmonics of Light-Induced Volume Gratingsin PMMA with Cold Neutrons

Frank HavermeyerFachbereich Physik, Universität Osnabrück, D-49069 Osnabrück, Germany

Sergei F. LyuksyutovInstitute of Physics, Ukrainian Academy of Science, 252650 Kiev, Ukraine

Romano A. RuppInstitut für Experimentalphysik, Universität Wien, A-1090 Wien, Austria

Helmut Eckerlebe, Peter Staron, and Jürgen VollbrandtGKSS Forschungszentrum, WFN, Geesthacht, Germany

(Received 12 August 1997)

Illumination of photosensitized (PMMA) with light triggers nonlinear polymerization processes. Inthis way periodical density structures with nearly104 linesymm can be generated which are beyondthe resolution limit of light-optical investigation methods. We report on the first direct experimentalobservation of such higher harmonics in deuterated (PMMA) by the nondestructive method of diffractionof neutrons at a wavelength of 1.2 nm. Diffraction efficiency of the second harmonic was found to be200–500 times weaker than that of the first order. The origin of higher harmonics is attributed to thenonlinearities of the processes of chain growth and of termination. [S0031-9007(98)05599-9]

PACS numbers: 61.12.–q, 42.40.– i, 42.65.–k, 42.70.–a

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One of the main advantages of small angle neutroscattering is that both organic (e.g., the structuresribosomes [1]) and inorganic (e.g., the location of clustein metals [2]) materials can be studied nondestructiveIn this context the discovery of neutron diffraction byphotoinduced gratings in polymers [3] has opened seveinteresting perspectives for polymer research: Diffractioof cold neutrons (with wavelengths from 0.5 to 1.5 nmprovides a very sensitive tool for the study of relaxatioprocesses in nonordered materials including thosepolymers [4] and can, in particular, be applied to the studof light-induced processes in polymers.

If PMMA, doped with a photoinitiator, is exposed toa sinusoidally modulated light pattern with periodL .2pyK, a mass density grating is created by polymeriztion which modulates the refractive index for neutron[5]. Besides the first harmonics such density gratingmay also exhibit higher harmonics with spatial frequencies 2K , 3K , . . . , hK. This can intuitively be understoodfrom the inherent nonlinearities of chain growth and otermination of the photoinitiated macroradicals which arinvolved in the formation of the density profile. Themodulation of the total number densityN , which includesfree monomers as well as monomer units bound in polmer chains, is given bydNsxd . dN0 1 dN1 cossKxd 1

dN2 coss2Kxd 1 · · · . The amplitude of the refractive in-dex modulation for neutrons is proportional todN. Itshth Fourier coefficient (h 0, 1, . . .) can be written as[5] dnh . sl2

ny2pd b̄c dNh, whereln is the wavelengthof the neutrons and̄bc is the coherent scattering lengthof one monomer unit (95 fm ford-PMMA). As it is

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possible to reachdN1 ø 1024 m23, which corresponds todn1 ø 1028, gratings can be produced which diffract netrons with an efficiency of about 71% [6]. An importanadvantage of PMMA is its high resolution capability whicis, e.g., required for the development of interferometerscold neutrons [7]. Artificial development of density graings with a profile containing strong higher harmonics pemits deflection of neutrons at even larger angles and coappreciably facilitate the development of neutron opticelements. The first experimental demonstration of suhigher harmonics of density gratings, which, in principlcannot be studied by the diffraction of light even thougthey were induced by light, and the physics of their origare the subjects of the current Letter.

The preparation of photosensitived-PMMA samplesstarts with purification of liquid d-MMA monomer(C5O2D8) with a content of 99.8% deuterium. A thermainitiator (to speed up prepolymerization) and a photinitiator with maximum absorption near 340 nm (tsensitize the samples in the near UV region) are addAll procedures were performed in an argon atmosphto avoid contact of the distilled monomer with air. Thprocess of prepolymerization ofd-PMMA took place for48 h at a temperature of 50±C.

Volume gratings with grating spacings of 400, 25and 204 nm were recorded at room temperature ithree d-PMMA samples using a conventional two-wavmixing configuration [8] and an Ar1 laser at a wavelengthof 351 nm. The laser beam was expanded 20 timesfully illuminate the whole area of each sample at a siof 400 mm2. However, only half of the sample wa

© 1998 The American Physical Society


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illuminated during recording of the grating withL 204 nm because the angle between the writing beamexceeded 120±. Total intensity of light on the entranceface of the sample was 150 Wym2, and the recording timewas 30 s. The diffraction efficiencies of the samples wedetermined at 351 nm by diffraction of beams attenuatby a factor of 100 with respect to the writing intensityFor all three samples, diffraction efficiency for lighexceeded 50% at the Bragg angle.

The neutron diffraction experiments were performedthe Small Angle Neutron Scattering Facility SANS-2 othe Geesthacht Neutron Facility (GeNF). A neutron beaat a wavelength of 1.2 nm with a wavelength distributioDlyl . 0.2 and a lateral divergence of 1.25 mrad waused in our experiments. The distance between tsample and the detector matrix was 21.72 m. The pixsize of the detector matrix was0.45 3 0.45 mm2. Thecross section was limited to10 3 10 mm2 by a Cddiaphragm in front of the sample. The neutron fluincident on the entrance face of the sample was1.45 3

104 s21. Only 1.08 3 104 neutrons per second weretransmitted by thed-PMMA samples. After correction forthe transmission of the quartz windows, this corresponto a transmission of 74%. The thickness of the sampled . 2.2 mm which gives a value of0.14 6 0.01 mm21

for the primary extinction coefficient.The diffraction patterns obtained with cold neutron

were particularly interesting for the samples with 250 an204 nm: distinct second order Bragg peaks correspondto spacings of125 6 5 nm and102 6 5 nm were clearlydetected for the samples with fundamental spacings of 2and 204 nm, respectively (Fig. 1). The numbersA1 andA2 of neutrons counted by the matrix detector in the spocorresponding to the spatial frequenciesK and 2K andthe numberAout of neutrons transmitted by the samplewere corrected by subtracting the statistically distributebackgrounds. From these data the diffraction efficiencih1,2sud A1,2sudyAout were calculated as a function of theangleu between the wave vector of the neutron flux anthe vector normal to the entrance face of the sample. Tneutrons in the2K spots were counted for 5400 and 3600for the samples with 204 and 250 nm, respectively, at eaangular setting. The diffraction efficiency of the seconharmonics as a function of the angle is shown in Fig.for all three samples. The diffraction orders6K have anexactly symmetrical location of the peaks and appear wequal magnitude and the same profiles. The widths of tBragg peaks of the first diffraction order (not shown iFig. 2) are additionally broadened byDu . 0.2u whichis, e.g., 0.48 mrad forL 204 nm because of the largedispersion of cold neutrons and by 1.25 mrad becausethe angular divergence of the neutron beam. Such larcontributions cause the effect of “shifting” of the diffractedspot on the detector matrix when readout is performed wchange of the angle. In order to normalize the data wirespect to the large spatial and temporal incoherence of

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FIG. 1. Distribution of the neutrons on the detector matrBragg spots corresponding to the second (2K) harmonics of thelight-induced density grating are clearly observed.

neutron beam, the “ideal” diffraction efficiency for a pefect neutron beamhid . d

L

R1`

2` hsud du was evaluatedby integrating the rocking curves [9]. Fromhid

1 , abso-lute values of the first harmonic of the density amplitudN1 of the monomer units were found to bes12 6 2d 3

1024 m23, s2.3 6 0.1d 3 1024 m23, and s1.6 6 0.2d 3

1024 m23 for 400, 250, and 204 nm, respectively. Thvalues for diffraction efficiencies of the second harmonwere h

id2 . 2.3 3 1022, 1.9 3 1023, and 5 3 1024 re-

sulting in dN2 s5 6 1d 3 1023 m23, s1.42 6 0.05d 3

1023 m23, and s0.73 6 0.02d 3 1023 m23 for 200, 125,and 102 nm, respectively. The origin of the experimetally detected second harmonics can be traced back tofundamental nonlinearity of the light-induced postpolmerization process ofd-PMMA. This process starts withthe decay of a molecule of the photoinitiatorS with aprobability ksI into two radical moleculesRp

0 under the

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FIG. 2. Diffraction efficiency of the second harmonics of thdensity grating ind-PMMA as a function of the angle ofincidence.

action of light with intensityI during exposure timet,

SksI! 2Rp

0 . (1)

The radicals grow by adding free monomers withprobability kp to the chain. This process results inthe creation of macroradicalsRp

n with different lengthsdepending on the number of monomer unitsn in theradical chain,

Rpn 1 M

kp! Rp

n11, n 0, 1 . . . . (2)

We assume here that the probabilities of the chemicalactions are independent from the chain length of the racal which is plausible above a certain chain length. By thcoupling of the two free radical electrons the interactinradicals can recombine to one polymer molecule with thsum of the monomer units [recombination of radicals—Eq. (3)]. The second process of termination is a hydrgen transfer reaction and results in two polymer molecul[disproportionation—Eq. (4)],

Rpn 1 Rp

mkt! Pn1m , (3)

Rpn 1 Rp

mkt! Pn 1 Pm . (4)

For simplicity, we assume here that the probabilitykt

of termination by recombination and disproportionatiois the same, and we do not distinguish between radicand polymer chains of different lengths, therefore introducing the total number density:NR

P`n0 NRn of radi-

cals. Now we are able to write the kinetic equationconsidering a one-dimensional model which describes t

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four main processes: reaction diffusion of the radical position, diffusion of monomer units, chain growth, andtermination of the radicals. Reaction diffusion of radi-cals is the migration of thes electrons during chaingrowth while the diffusion of monomers is a randomwalk of the monomer molecules. The current density omonomer units and of the radical molecules can be writen asJM,R . 2DM,RdNM,Rydx. HereDM and DR arediffusion coefficients for monomers and the reaction diffusion constants for radical (chains). By using the contnuity equation for monomers and radicals, we obtain thfollowing equations for the number density of radicalsNR

and monomersNM :

2DR≠2NR

≠x21 ktN

2R 2

≠NR

≠t, (5)

2DM≠2NM

≠x2 1 kpNRNM 2≠NM

≠t. (6)

A similar model has been studied by Marotz [10]However, the appearance of higher harmonics in thmodel was entirely attributed to the consequenceoverexposure in the photopolymerization process. Thdoes not apply to our experiments, because the expostime was so short that such a source of nonlinearity cabe excluded. In our model the nonlinearity is related tthe chemical processes during postpolymerization suchtermination of radicals [Eq. (5)].

After a brief exposure to light, the initial distribu-tion of radicals at the timet 0 can be written asNRs0d N̄Rs0d f1 1 a cossKxdg, wherea , 1, andN̄R isthe average of the initial distribution of radicals. Withthis starting condition, we solved Eqs. (5) and (6) numercally. By taking the values of the chemical constants foPMMA from the literature [11], we originally studied thebehavior up to the second harmonics, consideringdNM 12 dNM,0 1 dNM,1 cossKxd 1 dNM,2 coss2Kxd. This as-sumption is based on a Fourier analysis of the function NM and also on the experimental observation thaonly two harmonics,K and 2K, of the density grat-ing are detectable in PMMA. To fit our experimen-tal data with calculations, we selected the following seof parameters:kp 0.57 3 10231 m3 s21, kt 0.51 3

10228 m3 s21, and N̄M,0 11.88 3 1026 m23. We cal-culated the starting average value of the density of radicaby using experimental values of the absorption coefficieand the concentration6 3 1023 m23 of the photoinitiator.The time of postpolymerization was2.6 3 106 s and thecontrasta of the grating was 0.9.

However, to obtain coincidence between experimenand numerical calculations we used a value ofkp whichwas approximately 35 times larger than that in [11]We found that for a broad range of diffusion constant(DM , DR ø 10223 m2 s21), the effect of diffusion on thesolution forNM can be neglected. Curve No. 2 of Fig. 3shows a numerical solution forNM including the second


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FIG. 3. The number density profile for the monomer concetration. Curve No. 1: analytical solution with the presence oall harmonics; curves No. 2 and No. 3: numerical solution fothe first two and four harmonics, respectively, including the efect of diffusion.

harmonics. For comparison, curve No. 3 shows the effeof additional contributions up to the fourth harmonics.

The solutionNMstd NMs0d f1 1 NRs0dkttg2kpykt ofEqs. (5) and (6) for the case of neglected diffusion cabe obtained by straightforward calculation. HereNMs0dand NRs0d denote the initial concentrations for monomeand radical molecules. This solution is shown in Fig.for the selected values of the chemical constants as cuNo. 1. As can be seen, the effect of the third and fourharmonics are still needed to obtain good coincidenbetween numerical and analytical solutions. The fact thwe have not yet experimentally observed the compone3K and 4K can be explained by the fact that thediffraction efficiencyhh ~ dN2

M,h (for small efficiencies)decreases quadratically with thehth harmonicsdNM,h ofthe density modulation.

The limiting caseskp ¿ kt and kt ¿ kp of thissolution approachNMstd NMs0d exp

£2kpNRs0dt

§and

NMstd NMs0d h1 2 skpyktd lnf1 1 ktNRs0dtgj, respec-tively. In the first case, the nonlinearity is a consequenof the chain growth nonlinearitys~NRNmd and in thesecond one of the chain termination nonlinearitys~N2

Rd.From the considerations presented so far, we expect tvaluable information on the photopolymerization procesmight be obtained in future experiments by simultaneomeasurements of the time dependence of the first and sond order diffraction with light and neutron waves fromlight-induced gratings.

Light absorption of deuterated PMMA doped withphotoinitiator steeply rises below 320 nm, where threfractive index attains the value ofn 1.523 [12].Therefore gratings with spacings belowL ly2n 105 nm cannot, on principle, be detected by ligh

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diffraction experiments even at the limiting wavelengt320 nm. If the sample shape and/or the direction of thgrating vector only permits readout under transmissio(Laue) conditions, then the practical limit is even highenamely,L ly2 160 nm. Therefore our experimentdemonstrates for the first time that gratings arising frolight-triggered nonlinear processes can be studied by netron diffraction methods, when they cannot, in principlebe detected by optical means because of the onset offundamental optical absorption. By this we establishea new technique for the study of nonlinear processwhich are needed for the nanostructurization of mattea topic which is of high importance, e.g., for the desigof interferometers for cold and ultracold neutrons [7]. Inaddition, we want to stress that our method clearly allowus to distinguish between linear and nonlinear processwith first and second order kinetics, respectively. Wobtained clear evidence that nonlinear chemical procesplay an important role in the photopolymerization dowto a spatial frequencyK 3.08 3 1022 nm21.

In summary, we experimentally observed a profile othe density grating ind-PMMA with a characteristicperiod near 100 nm. It is not possible to resolve sucholographic structures with existing optical methods. Tour knowledge, this is the first report on the possibilitof directly observing higher harmonics of density gratingin polymer materials at a primary grating spacing of lesthan 300 nm.

This work was funded by the BMBF of the FRG(Project No. RU11.12K) and supported by the GKSS.

[1] H. B. Stuhrmann, Phys. Scr.T49, 644 (1993).[2] G. Kostorz, J. Appl. Cryst.24, 444 (1991).[3] R. A. Rupp, J. Hehmann, R. Matull, and K. Ibel, Phys

Rev. Lett.64, 301 (1990).[4] R. Matull, P. Eschkötter, R. A. Rupp, and K. Ibel,

Europhys. Lett.15, 133 (1991).[5] H. Dachs,Topics in Current Physics—Neutron Diffrac-

tion (Springer-Verlag, Berlin, 1978).[6] R. A. Rupp, R. Heintzmann, S. Breer, and U. Schellhorn

Institute Laue Langevin Experimental Report No. 5-16237, 1995.

[7] U. Schellhorn, R. A. Rupp, S. Breer, and R. P. MayPhysica (Amsterdam)234–236B, 1068 (1997).

[8] N. Kukhtarev, V. Markov, S. Odulov, M. Soskin, andV. Vinetskii, Ferroelectrics22, 949 (1979).

[9] R. A. Rupp, U. Hellwig, U. Schellhorn, J. Kohlbrecher,A. Wiedenmann, and J. Woods, Polymer10, 2299(1997); see also U. Schellhorn, Dissertation, Universitof Osnabrück, 1997.

[10] J. Marotz, Appl. Phys. B37, 181 (1985).[11] H.-J. Cantow and M. Stickler, Makromol. Chem.184,

2563 (1983).[12] F. Mesegner and C. Sanchez, J. Mater. Sci.15, 53 (1980).

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nondestructive resolution of higher harmonics of light...

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